<p><b>Abstract</b>—A number of applications in computer-aided manufacturing, CAD, and computer-aided geometric design ask for triangulating pieces of material with defects. These tasks are known collectively as <it>constrained triangulations</it>. Recently, a powerful architecture called the reconfigurable mesh has been proposed: In essence, a reconfigurable mesh consists of a mesh-connected architecture augmented by a dynamically reconfigurable bus system. The main contribution of this paper is to show that the flexibility of the reconfigurable mesh can be exploited for the purpose of obtaining constant-time algorithms for a number of constrained triangulation problems. These include triangulating a convex planar region containing any constant number of convex holes, triangulating a convex planar region in the presence of a collection of rectangular holes, and triangulating a set of ordered line segments. Specifically, with a collection of O(<it>n</it>) such objects as input, our algorithms run in O(1) time on a reconfigurable mesh of size <it>n</it>×<it>n</it>. To the best of our knowledge, this is the first time constant time solutions to constrained triangulations are reported on this architecture.</p>